Controlled MCMC for Automatic Sampler Calibration
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Christophe Andrieu, University of Bristol 
(joint work with C.P. Robert and E. Moulines)


In this talk we present an original and general framework for automatically optimising
the statistical properties of Markov chain Monte Carlo (MCMC) samplers.

Classical MCMC samplers usually depend on parameters, say \theta, that need to be tuned
in order to lead to efficient algorithms. It is well known for example that the
performance of a Metropolis-Hastings algorithm will heavily depend on the choice of a
`good' proposal distribution, which might depend on some parameters. A natural goal is
therefore to optimise the set of parameters \theta, on which the sampler depends, in
order to satisfy some statistical criteria.

The methodology we propose allows for the self-tuning of the Markov chain process in the
light of its history: the sampler therefore learns ``on the fly'' the optimal set of 
parameters.

The method is supported by theoretical results which prove the convergence of the method
under fairly general and applicable conditions. A particular emphasis is given on
non-asymptotic and explicit bounds on the convergence of ergodic averages. These
bounds have clear practical interpretations in terms of basic properties of the transition
probability of the Markov chain and give some insight into the potential benefits of
such adaptation schemes.

We present several detailed examples of applications and numerical experiments. In
particular we show how optimal blocking can be chosen in the context of stochastic
volatility models (cf Shephard & Pitt, 1997).